mesh breaking apart and recombining to form another mesh?
Another way is morphing objects with Particle Systems



You can morph shapes of objects and even objects themselves (influnced by Explode modifier) with Keyed Particle Systems.

From wiki:


  The particle paths of keyed particles are determined from the emitter to another particle system’s particles. This allows creation of chains of systems with keyed physics to create long strands or groovy moving particles. 


You should only setup them in appropriate way so to make particles travel from one emitter to another.

Step 1: Add objects and assign "simple" particles systems

These systems don't do anything but providing particles needed for action. Because we need them to not emit in time, setup this system (as you can link to both objects the same) so to change Number of particles, Start and End time, Lifetime, Velocity.

Step 2: Add Keyed particle system 

Add 2 different Keyed Particle systems - 1 per object. As in the gif above, first setup object on the left. Change Number, Start and End, Lifetime to the same values as for the "simple" particle system and set Velocity > Normal to something more than 1.

And the main, in Physics scroll select Keyed.  Add 2 keyes (in
    both setups) and setup them in this way:


for object Sphere: 
    - 1-st key: second ParticleSystem of the Suzanne (use slider in the bottom to select Particle System of the object);
    - 2-nd key: second ParticleSystem of the Sphere.
for object Suzanne:
      - 1-st key: second ParticleSystem of the Sphere;
      - 2-nd key: second ParticleSystem of the Suzanne.


(in example above I also changed timing in Time and Duration sliders so to make systems not interfere with each other, and materials transparency, in order not to see already broken object etc.)

Now, for both objects add Explode modifier, and move it to the bottom (under both particle systems). Choose Cut Edges for finer result.



Note that this way doesn't provide accurate result and it could be greatly enhanced. Particulalry, here are some disadvantages:


Objects' topology means a lot for this to work. Particles will be emitted taking into account amount and distribution of them on the faces.
Keyed Particles aren't affected by physics (so force fields, gravity and so on). Read more about it in this related question. In order to further improve such animation you may consider using 2 approaches:


using simple systems as keys in Keyed Particle System setup (it would involve a lot of cheating to make particles form initial objects)
using 3-rd object and 3-rd Keyed Particle system which shold represent "falling particles down" (if simulating gravity).



Although it could be a beginning for some effects. 



(note that stripdown version of file is linked; it includes morphing objects, their setup, ground plane and light).



Credits for particle systems setup:


very good starting lesson on keyed particle
systems